Is it possible to add distance constraints to the vertices of a mesh?
Doing something like that manually would be relatively hard, but I would recommend using proportional editing (O, Scroll Wheel to adjust strength). That should be able to do most of it. The next best option would be sculpting. Try using the 'grab brush' to get the overall shape, then just play around with various brushes like the clay brush or the draw brush to get some smaller details, just fiddle with those until you find what you like. Try taking a look at this and this. One of them is a tutorial for animating, you probably won't find much specifically on modelling the paper, but you might find some things in there to do with how he modelled it, and the other short one is just an animation, you might have to do a little bit of digging around to find out how he did it if that's along the lines of what you are doing.